Lorna Gregory (Università degli Studi della Campania Luigi Vanvitelli)

Decidability of Theories of Modules of Prüfer domains

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An integral domain is Prüfer if its localisation at each maximal ideal is a valuation domain. Many classically important rings are Prüfer domains. For instance, they include Dedekind domains and hence rings of integers of number fields; Bézout domains and hence the ring of complex entire functions and the ring of algebraic integers; the ring of integer valued polynomials with rational coefficients and the real holomorphy rings of formally real fields.

Over the last 15 years, efforts have been made to characterise when the theory of modules of (particular types of) Prüfer domains are decidable. I will give an overview of such decidability results culminating in recently obtained elementary conditions completely characterising when the theory of modules of an arbitrary Prüfer domain is decidable.